This appendix is a quick introduction to locally presentable categories. This notion is in some sense a formalization of what is an algebraic structure. When category theory is restricted to locally presentable categories, many things get simpler. In particular, there are characterizations of adjoint functors purely in terms of preservation of limits and colimits. Locally presentable categories also play an important role in the theory of model categories through the concept of combinatorial model categories. There are many ways to define locally presentable categories. The appendix begins by presenting the concept using sketches, which encode the syntax of an algebraic structure. These sketches are used several times in the body of the book. The intrinsic categorical characterization is then provided, introducing several notions that are important for the theory of model categories. Finally, the syntactic characterization is discussed.

Among the many existing notions of higher categories, the notion of strict globular n-category is, in some sense, the most basic one. In this chapter, the essential definitions and notations are set. Starting with a description of the basic "shapes", that is, the presheaf category of globular sets, family of operations endowing a globular set with a structure of ω-category is defined. Then, it is proven that the category of strict ω-categories is exactly the category of algebras of the monad induced by the forgetful functor from ω-categories to globular sets. Finally, important subcategories of ω-categories, obtained by requiring cells to be invertible above a given dimension, are defined.

The notion of polygraph introduced so far is a particular case of a general construction due to Batanin. In fact, any finitary monad on globular sets yields a appropriate notion of polygraph. The original motivation was the study of weak ω-categories seen as algebras of such a monad. Another example, of particular relevance to this book, is the case of linear polygraphs presented in the last section.

This chapter discusses 1-polygraphs, which are simply directed graphs, thought of here as abstract rewriting systems: they consist of vertices, which represent the objects of interest, and arrows, which indicate that one object can be rewritten into another. After formally introducing those, it will be shown that they provide a notion of presentation for sets, by generators and relations. Of course presentations of sets are of little interest in themselves, but merely used here as a gentle introduction to some of the main concepts discussed in this work: in particular, the notion of Tietze transformations is introduced, which generates the equivalence between two presentations of the same set. In this context, an important question consists in deciding when two objects are equivalent, i.e., represent the same element of the presented set. In order to address it, the theory of abstract rewriting systems is developed.

The usefulness and richness of 2-polygraphs is confirmed by the large number and variety of categories they present. In order to show that a given polygraph is a presentation of a given category, one can either tackle the issue directly, by using rewriting tools, or take a modular approach, by combining already known presentations: this is the route taken in the present chapter. Three significant applications are given. First addressed is the presentation of limits and colimits by means of given presentations of the base categories, and precisely shown is how to systematically build presentations of products, coproducts, and pushouts. Next, it is shown how to add formal inverses to some morphisms of a category at the level of presentations. Finally, distributive laws are investigated in relation to factorization systems on categories. A notion of composition along a distributive law between two small categories sharing the same set of objects is introduced, and it is shown how to derive a presentation of this composite from presentations of the components.

The present chapter concentrates on some useful families of n-polygraphs based on familiar shapes: cylinders, cubes, and simplices. These families are crucial in the development of a homotopy theory of ω-categories. Two methods for generating these families are explained. The first one is based on a direct definition of the cylinder polygraph of a polygraph. The second is based on Steiner’s theory of augmented directed complexes, which is a very powerful tool to build polygraphs using chain complexes. In particular, it allows to define a tensor product for polygraphs (or even ω-categories) from which the cylinder polygraph, as well as a join operation, can be recovered.

This chapter presents rewriting techniques for associative algebras. Here, algorithms are sought to turn a given presentation by generators and relations into a rewriting system by orienting the latter, thereby producing linear bases of the presented algebra. In particular, this approach applies to various fundamental decision problems, such as the word problem, ideal membership, or to compute quadratic bases, e.g., Poincaré-Birkhoff-Witt bases, Hilbert series, syzygies of presentations, homology groups, and Poincaré series. However, if rewriting rules are required to be compatible with the linear structure, an immediate problem arises: no rewriting system can be terminating. In order to fix this problem, the structure of linear polygraph with an appropriate notion of reduction can be considered. Linear polygraphs are introduced as a framework for linear rewriting, their confluence properties are studied, and Gröbner bases and Poincaré-Birkhoff-Witt bases are expressed in the setting of linear polygraphs.

This chapter establishes 3-polygraphs as a notion of presentation for 2-categories. As expected, those consist in generators for 0-, 1- and 2-dimensional cells, together with relations between freely generated 2-cells, which are represented by generating 3-cells. Any 3-polygraph induces an abstract rewriting system, so that all associated general rewriting concepts make sense in this setting: confluence, termination, etc. However, more specific tools have to be adapted to this context: the notion of critical branching is defined here for 3-polygraphs, along with the proof that confluence of critical branchings implies the local confluence of the polygraph. In the case where the polygraph is terminating, local confluence implies confluence, providing a systematic method to show the convergence of a 3-polygraph. When this is the case, normal forms give canonical representatives for 2-cells modulo the congruence generated by 3-cells, and it is explained how to exploit this to show that a given 3-polygraph is a presentation of a given 2-category.

This appendix presents examples of coherent presentations of monoids. In particular, focus is placed on families of monoids which occur in algebra and whose coherent presentations are computed using the rewriting method that extends Squier’s and Knuth-Bendix’s completion procedures into a homotopical completion-reduction procedure. Coherent presentations of monoids are shown to explicitly describe the actions of monoids on small categories. This construction is applied to the case of Artin monoids. In particular, it is proven that the Zamolodchikov 3-generators extend the Artin presentation into a coherent presentation and, as a byproduct, a constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category is given. Coherent presentations of plactic and Chinese monoids are also provided.

This chapter introduces all the notions and tools necessary to define and establish the existence of the folk model category structure on ω-categories. Particular focus is placed on the concept of ω-equivalences, which will serve as the weak equivalences of this model structure. The class of ω-equivalences is the appropriate generalization to ω-categories of the class of equivalences of ordinary categories. In particular, an ω-equivalence between 1-categories is nothing but an equivalence of categories. To define this notion, it is necessary to generalize the concept of an invertible cell (or isomorphism). This leads to the notion of a reversible cell, which is, in intuitive terms, a cell admitting an inverse up to cells admitting inverses, up to cells admitting inverses, etc. Another fundamental tool is the ω-category of reversible cylinders in an ω-category, which will lead to a sensible notion of homotopy.

This appendix recalls basic notions on modules, abelian resolutions, and homology.

This chapter recasts the notion of string rewriting system into the language of polygraphs. This notion, which consists of a set of pairs of words called relations or rewriting rules over a fixed alphabet, is introduced along with a more general variant adapted to categories. It is shown that the rewriting paths form the morphisms of a sesquicategory, in which the traditional concepts for abstract rewriting systems can be instantiated. The word problem is then introduced, and it is shown that it can be efficiently solved for convergent, i.e., confluent and terminating rewriting systems. In practice, confluence can be checked by inspecting the critical branchings of the rewriting system, and termination by introducing a suitable reduction order. The convergence of a rewriting system is also useful to show that it forms a presentation of a given category. Finally, residuation techniques are introduced, which allow proving useful properties of categories (such as the existence of pushouts) by performing computations on their presentations.

This chapter establishes the main properties of the category of n-polygraphs. Limits and colimits are computed, and the category is proven to be complete and cocomplete. The behavior of the cartesian product deserves a special attention in that it does not correspond to the product of generators. The monomorphisms (resp. epimorphisms) in are then characterized as injective (resp. surjective) maps between generators. The linearization of polygraphic expressions plays a central role in proving these facts. Whereas the category of n-polygraph is a presheaf category in low dimensions, it already fails to be cartesian closed for n=3, the culprit for this defect being as usual the Eckmann-Hilton phenomenon. The categories of n-polygraph are, however, locally presentable. The technical notion of context is introduced in relation with n-dimensional rewriting, and used to prove that if an ω-category is freely generated by a polygraph, then this polygraph is unique up to isomorphism. Finally, rewriting properties of n-polygraphs are defined and coherence results are proven by rewriting on (n-1)-categories presented by convergent n-polygraphs.